point group
/ poynt groop /
Hold a single object — say a six-sided die — fixed at its center and ask: in how many ways can I turn or mirror it so it looks exactly the same? You can spin it about several axes and flip it across several mirrors, and all those moves share one feature: they leave the very center untouched. That complete set of moves is what mathematicians call a point group.
A point group is the set of all symmetry operations — rotations, reflections, and inversions — that leave a crystal looking unchanged while keeping at least one point fixed in place. The fixed point is the catch: point-group operations do not slide the crystal around, only rotate and mirror it about that anchor. For ordinary periodic crystals, the no-five-fold restriction means there are exactly 32 distinct crystallographic point groups, no more and no less, and every crystal's overall shape and directional properties belong to one of them.
Point groups matter because they govern a material's directional, shape-related behavior: whether it can be piezoelectric, how its optical or elastic response varies with direction, and what its outward crystal faces can look like. The key distinction to hold onto is between the point group, which ignores translations and concerns only the orientation symmetries about a point, and the fuller space group, which adds the sliding symmetries of the repeating lattice. The point group is the orientation part; the space group is the whole story.
A cube and a sphere differ in point group: rotate a sphere any tiny amount and it looks the same, but a cube only looks the same after quarter-turns about its faces and other special rotations. Listing exactly those allowed moves names the cube's point group.
A cube's allowed turns and mirrors, listed together, are its point group.
There are exactly 32 crystallographic point groups for periodic crystals — a finite, complete list. Free molecules can have point groups outside this 32 (including five-fold ones), because a single molecule need not tile space.